SERVE-Boundary Conditions
305
G ij (x, x
F I ) =
1
4πμ
δ ij
r I −
1
4 (1 − ν)
r
I
,ij
(16)
where r I = |x − x F I | is the separation distance between a source point x F I and a
generic field point x.
Closed form expressions for the integrals in equations (14) and (15) have been
derived using elliptic integrals [42] with spatially invariant eigenstrains inside
ellipsoidal inclusions. The perturbed strains due to any isolated (noninteracting)
inclusion F I in the MVE are expressed as:
∗
ij (x) =
mve
H ij kl
x, ˆ
x
kl
ˆ
x
d ˆ
x
(17)
where ˆ
x is a point in the inclusion. H ij kl corresponds to a unified 2-point Eshelby
tensor given as:
H ij kl
x, ˆ
x
= ι
F I (x) S
F I
ij kl +
1 − ι
F I (x)
ˆ
G
F I
ij kl
x, ˆ
x
(18)
where S
F I
ij kl and ˆ
G
F I
ij kl
x, ˆ
x
are the interior and exterior Eshelby tensors. The
corresponding perturbed displacements are written in terms of the Eshelby tensors
as:
u
∗
i (x) =
mve
L ikl
x, ˆ
x
kl
ˆ
x
d ˆ
x
(19)
where
L ikl
x, ˆ
x
= ι
F I (x) T
F I
ikl
x, ˆ
x
+
1 − ι
F I (x)
D
F I
ikl
x, ˆ
x
(20)
Expression for the interior and exterior Eshelby tensors S
F I
ij kl and ˆ
G
F I
ij kl
x, ˆ
x
, as
well as the displacement-transfer tensors T
F I
ikl
x, ˆ
x
and D
F I
ikl
x, ˆ
x
for a circular
cylindrical fiber are given in the Appendix. For identical fibers in mve , the
following reductions hold:
S
F I
ij kl = S
F J
ij kl = S ij kl
M ij kl
x
I
= M ij kl
x
J
= M ij kl
ˆ
G
F I
ij kl (r) = ˆ
G
F J
ij kl (r) = ˆ
G ij kl (r, θ )
where S ij kl and M ij kl are spatially invariant, and ˆ
G ij kl is position dependent and
describes interactions between fibers.